a new equilibrated residual method: improving accuracy and …s/2017_cmn17_slides.pdf · 2017. 7....
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A new equilibrated residual method:improving accuracy and efficiency of
flux-free error estimates
Nuria PARES and Pedro DIEZ
Laboratori de Calcul Numeric (LaCaN)Universitat Politecnica de Catalunya (Barcelona)
http://www-lacan.upc.edu
A new equilibrated residual method (CMN 2017) - July 2017 - 1/21
Certification of FE approximationsThe finite element method is a basic tool in engineering designand since engineering decisions are based on approximations ofthe solution, it is crucial to certify the quality of the results.
GOAL: provide a guaranteed interval for thevalue of the QoI
`O(u) =
∫Γ
σVM(u)dΓ
`O(u) ∈ [L−, L+]
HOW?
1. compute `O(uh)
2. define `O(e) = `O(u)− `O(uh)
3. bound s− ≤ `O(e) ≤ s+
`O(u) ∈ [`O(uh) + s−, `O(uh) + s+]
We need a posteriori GUARANTEED/STRICT error bounds.They also have to be ACCURATE and CHEAP
A new equilibrated residual method (CMN 2017) - July 2017 - 2/21
Certification of FE approximationsThe finite element method is a basic tool in engineering designand since engineering decisions are based on approximations ofthe solution, it is crucial to certify the quality of the results.
GOAL: provide a guaranteed interval for thevalue of the QoI
`O(u) =
∫Γ
σVM(u)dΓ
HOW?
1. compute `O(uh)
2. define `O(e) = `O(u)− `O(uh)
3. bound s− ≤ `O(e) ≤ s+
`O(u) ∈ [`O(uh) + s−, `O(uh) + s+]
We need a posteriori GUARANTEED/STRICT error bounds.They also have to be ACCURATE and CHEAP
A new equilibrated residual method (CMN 2017) - July 2017 - 2/21
Certification of FE approximationsThe finite element method is a basic tool in engineering designand since engineering decisions are based on approximations ofthe solution, it is crucial to certify the quality of the results.
GOAL: provide a guaranteed interval for thevalue of the QoI
`O(u) =
∫Γ
σVM(u)dΓ
HOW?
1. compute `O(uh)
2. define `O(e) = `O(u)− `O(uh)
3. bound s− ≤ `O(e) ≤ s+
`O(u) ∈ [`O(uh) + s−, `O(uh) + s+]
We need a posteriori GUARANTEED/STRICT error bounds.
They also have to be ACCURATE and CHEAP
A new equilibrated residual method (CMN 2017) - July 2017 - 2/21
Certification of FE approximationsThe finite element method is a basic tool in engineering designand since engineering decisions are based on approximations ofthe solution, it is crucial to certify the quality of the results.
GOAL: provide a guaranteed interval for thevalue of the QoI
`O(u) =
∫Γ
σVM(u)dΓ
HOW?
1. compute `O(uh)
2. define `O(e) = `O(u)− `O(uh)
3. bound s− ≤ `O(e) ≤ s+
`O(u) ∈ [`O(uh) + s−, `O(uh) + s+]
We need a posteriori GUARANTEED/STRICT error bounds.They also have to be ACCURATE and CHEAP
A new equilibrated residual method (CMN 2017) - July 2017 - 2/21
Guaranteed, accurate and efficient bounds
CERTIFICATION
complementary energy + implicit error estimatorsdual formulation for the error involving only local problems
A new equilibrated residual method (CMN 2017) - July 2017 - 3/21
Guaranteed, accurate and efficient bounds
CERTIFICATION
complementary energy + implicit error estimatorsdual formulation for the error involving only local problems
CHEAPER
Hybrid-flux estimatorsequilibrated
MORE ACCURATE
Flux-free estimatorsstars/subdomain [PDH2006]
A new equilibrated residual method (CMN 2017) - July 2017 - 3/21
Guaranteed, accurate and efficient bounds
CERTIFICATION
complementary energy + implicit error estimatorsdual formulation for the error involving only local problems
CHEAPER
Hybrid-flux estimatorsequilibrated
MORE ACCURATE
Flux-free estimatorsstars/subdomain [PDH2006]
CHEAP + ACCURATE
EXPLICIT Flux-free estimator
A new equilibrated residual method (CMN 2017) - July 2017 - 3/21
Model problemReaction-diffusion equation: −∆u+ κ2u = f in Ω,
u = uD
on ΓD,∇u · n = g
Non ΓN.
Weak form: find u ∈ U such that∫Ω
(∇u ·∇v + κ2uv
)dΩ︸ ︷︷ ︸
a(u,v)
=
∫Ω
fv dΩ +
∫ΓN
gNv dΓ︸ ︷︷ ︸
`(v)
∀v ∈ V .
Finite element approximation: find uh ∈ Uh such that
a(uh, v) = `(v) for all v ∈ Vh.
Error equations: find e = u− uh ∈ V such that
a(e, v) = `(v)− a(uh, v) = R(v) for all v ∈ V .
triangular mesh + linear elements
A new equilibrated residual method (CMN 2017) - July 2017 - 4/21
Guaranteed error boundsThe complementary energy approach allows to overestimate |||e|||
approach introduced by Fraeijs de Veubeke in 1964
a(e, v) =
∫Ω
(∇e ·∇v + κ2ev
)dΩ = R(v) for all v ∈ V∫
Ω
(q ·∇v + κ2rv
)dΩ = R(v) for all v ∈ V
new error unknowns
Dual formulation:Any pair of dual estimates (q, r) such that∫
Ω
(q ·∇v + κ2rv
)dΩ = R(v) for all v ∈ V
provide an upper bound for the energy norm of the error
|||e|||2 =
∫Ω
(∇e ·∇e+ κ2e2
)dΩ ≤
∫Ω
(q · q + κ2r2
)dΩ
complementary energy
A new equilibrated residual method (CMN 2017) - July 2017 - 5/21
Guaranteed error boundsThe complementary energy approach allows to overestimate |||e|||
approach introduced by Fraeijs de Veubeke in 1964
a(e, v) =
∫Ω
(∇e ·∇v + κ2ev
)dΩ = R(v) for all v ∈ V∫
Ω
(q ·∇v + κ2rv
)dΩ = R(v) for all v ∈ V
new error unknowns
Dual formulation:Any pair of dual estimates (q, r) such that∫
Ω
(q ·∇v + κ2rv
)dΩ = R(v) for all v ∈ V
provide an upper bound for the energy norm of the error
|||e|||2 =
∫Ω
(∇e ·∇e+ κ2e2
)dΩ ≤
∫Ω
(q · q + κ2r2
)dΩ
complementary energyA new equilibrated residual method (CMN 2017) - July 2017 - 5/21
Guaranteed error bounds
Global problem: find q and r
s.t.
∫Ω
(q ·∇v + κ2rv
)dΩ = R(v) TOO EXPENSIVE
minimizing the upper bound
∫Ω
(q · q + κ2r2
)dΩ
SPLIT THE GLOBAL PROBLEM INTO LOCAL PROBLEMS
Hybrid-flux estimators Flux-free estimators
A new equilibrated residual method (CMN 2017) - July 2017 - 6/21
Guaranteed error bounds
Global problem: find q and r
s.t.
∫Ω
(q ·∇v + κ2rv
)dΩ = R(v) TOO EXPENSIVE
minimizing the upper bound
∫Ω
(q · q + κ2r2
)dΩ
SPLIT THE GLOBAL PROBLEM INTO LOCAL PROBLEMS
Hybrid-flux estimators Flux-free estimators
A new equilibrated residual method (CMN 2017) - July 2017 - 6/21
Guaranteed error bounds
Global problem: find q and r
s.t.
∫Ω
(q ·∇v + κ2rv
)dΩ = R(v) TOO EXPENSIVE
minimizing the upper bound
∫Ω
(q · q + κ2r2
)dΩ
SPLIT THE GLOBAL PROBLEM INTO LOCAL PROBLEMS
Hybrid-flux estimators Flux-free estimators
A new equilibrated residual method (CMN 2017) - July 2017 - 6/21
Guaranteed error bounds
Global problem∫Ω
(q ·∇v + κ2rv
)dΩ = R(v)
Hybrid-flux
∫Ωk
(qk ·∇v + κ2rkv
)dΩ = Rk(v)+
∫Ωk
gkv dΓ
single element q|Ωk = qk , r|Ωk = rk
Flux-free
∫ωi
(qi ·∇v + κ2riv
)dΩ = R(φiv)
patch of elements q =nnp∑i=1
qi , r =nnp∑i=1
ri
A new equilibrated residual method (CMN 2017) - July 2017 - 7/21
Guaranteed error bounds
Global problem∫Ω
(q ·∇v + κ2rv
)dΩ = R(v)
Hybrid-flux
∫Ωk
(qk ·∇v + κ2rkv
)dΩ = Rk(v)+
∫Ωk
gkv dΓ
single element q|Ωk = qk , r|Ωk = rk
Flux-free
∫ωi
(qi ·∇v + κ2riv
)dΩ = R(φiv)
patch of elements q =nnp∑i=1
qi , r =nnp∑i=1
ri
Ωk
A new equilibrated residual method (CMN 2017) - July 2017 - 7/21
Guaranteed error bounds
Global problem∫Ω
(q ·∇v + κ2rv
)dΩ = R(v)
Hybrid-flux
∫Ωk
(qk ·∇v + κ2rkv
)dΩ = Rk(v)+
∫Ωk
gkv dΓ
single element q|Ωk = qk , r|Ωk = rk
Flux-free
∫ωi
(qi ·∇v + κ2riv
)dΩ = R(φiv)
patch of elements q =nnp∑i=1
qi , r =nnp∑i=1
ri
ωi
A new equilibrated residual method (CMN 2017) - July 2017 - 7/21
Hybrid-flux / equilibrated error estimatesSTEP 1: loop in nodes to compute the equilibrated tractions gk
STEP 2: loop in elements tocompute the dual fluxes
(qk, rk)at each element Ωk
independently
equilibrated fluxes
A new equilibrated residual method (CMN 2017) - July 2017 - 8/21
Hybrid-flux / equilibrated error estimatesSTEP 1: loop in nodes to compute the equilibrated tractions gk
STEP 2: loop in elements tocompute the dual fluxes
(qk, rk)at each element Ωk
independently
equilibrated fluxes
A new equilibrated residual method (CMN 2017) - July 2017 - 8/21
Hybrid-flux / equilibrated error estimatesSTEP 1: loop in nodes to compute the equilibrated tractions gk
STEP 2: loop in elements tocompute the dual fluxes
(qk, rk)at each element Ωk
independently
equilibrated fluxes
A new equilibrated residual method (CMN 2017) - July 2017 - 8/21
Hybrid-flux / equilibrated error estimatesSTEP 1: loop in nodes to compute the equilibrated tractions gk
STEP 2: loop in elements tocompute the dual fluxes
(qk, rk)at each element Ωk
independently
equilibrated fluxes
A new equilibrated residual method (CMN 2017) - July 2017 - 8/21
Hybrid-flux / equilibrated error estimatesSTEP 1: loop in nodes to compute the equilibrated tractions gk
STEP 2: loop in elements tocompute the dual fluxes
(qk, rk)at each element Ωk
independently
equilibrated fluxes
A new equilibrated residual method (CMN 2017) - July 2017 - 8/21
Hybrid-flux / equilibrated error estimatesSTEP 1: loop in nodes to compute the equilibrated tractions gk
STEP 2: loop in elements tocompute the dual fluxes
(qk, rk)at each element Ωk
independently
equilibrated fluxes
A new equilibrated residual method (CMN 2017) - July 2017 - 8/21
Hybrid-flux / equilibrated error estimatesSTEP 1: loop in nodes to compute the equilibrated tractions gk
STEP 2: loop in elements tocompute the dual fluxes
(qk, rk)at each element Ωk
independently
equilibrated fluxes
A new equilibrated residual method (CMN 2017) - July 2017 - 8/21
Flux-free error estimatesSTEP 1: loop in nodes to compute the dual fluxes in the stars
(qi, ri) in ωi (patch of elements)
STEP 2: add all the local contributions and compute the norm
q =nnp∑i=1
qi , r =nnp∑i=1
ri
A new equilibrated residual method (CMN 2017) - July 2017 - 9/21
Flux-free error estimatesSTEP 1: loop in nodes to compute the dual fluxes in the stars
(qi, ri) in ωi (patch of elements)
STEP 2: add all the local contributions and compute the norm
q =nnp∑i=1
qi , r =nnp∑i=1
ri
A new equilibrated residual method (CMN 2017) - July 2017 - 9/21
Flux-free error estimatesSTEP 1: loop in nodes to compute the dual fluxes in the stars
(qi, ri) in ωi (patch of elements)
STEP 2: add all the local contributions and compute the norm
q =nnp∑i=1
qi , r =nnp∑i=1
ri
A new equilibrated residual method (CMN 2017) - July 2017 - 9/21
Flux-free error estimatesSTEP 1: loop in nodes to compute the dual fluxes in the stars
(qi, ri) in ωi (patch of elements)
STEP 2: add all the local contributions and compute the norm
q =nnp∑i=1
qi , r =nnp∑i=1
ri
A new equilibrated residual method (CMN 2017) - July 2017 - 9/21
Flux-free error estimatesSTEP 1: loop in nodes to compute the dual fluxes in the stars
(qi, ri) in ωi (patch of elements)
STEP 2: add all the local contributions and compute the norm
q =nnp∑i=1
qi , r =nnp∑i=1
ri
A new equilibrated residual method (CMN 2017) - July 2017 - 9/21
Flux-free error estimatesSTEP 1: loop in nodes to compute the dual fluxes in the stars
(qi, ri) in ωi (patch of elements)
STEP 2: add all the local contributions and compute the norm
q =nnp∑i=1
qi , r =nnp∑i=1
ri
A new equilibrated residual method (CMN 2017) - July 2017 - 9/21
Flux-free error estimatesSTEP 1: loop in nodes to compute the dual fluxes in the stars
(qi, ri) in ωi (patch of elements)
STEP 2: add all the local contributions and compute the norm
q =nnp∑i=1
qi , r =nnp∑i=1
ri
A new equilibrated residual method (CMN 2017) - July 2017 - 9/21
Computational cost overview
Equilibrated Flux-free
Lo
opon
node
s
DOF: DOF:
one per edge of ωi dof of (qik, rik)
× elements of ωi
Lo
opon
elem
ents
DOF:
dof of (qk, rk)
A new equilibrated residual method (CMN 2017) - July 2017 - 10/21
Computational cost overview
Equilibrated Flux-free
Lo
opon
node
s
DOF: DOF:
one per edge of ωi dof of (qik, rik)
× elements of ωi
Lo
opon
elem
ents
DOF: higher cost
dof of (qk, rk) better accuracy
A new equilibrated residual method (CMN 2017) - July 2017 - 10/21
Computational cost overview
Equilibrated Flux-free
Lo
opon
node
s
DOF: NEW!!!
one per edge of ωi two per edge of ωi
Lo
opon
elem
ents
DOF:
dof of (qk, rk)
A new equilibrated residual method (CMN 2017) - July 2017 - 10/21
New guaranteed, accurate and cheaperror estimate (EE)
Local problems: find qi and ri such that∫ωi
(qi ·∇v + κ2riv
)dΩ = R(φiv) ∀v ∈ V(ωi)
minimizing the local complementary energy∫ωi
(qi · qi + κ2(ri)2
)dΩ
ωi
φi
KEY POINT
Find a closed EXPLICIT solution for qi
A new equilibrated residual method (CMN 2017) - July 2017 - 11/21
New guaranteed, accurate and cheaperror estimate (EE)
Local problems: find qi such that∫ωi
qi ·∇vdΩ = R(φiv) ∀v ∈ V(ωi)
minimizing the local complementary energy∫ωi
qi · qidΩωi
φi
KEY POINT
Find a closed EXPLICIT solution for qi
A new equilibrated residual method (CMN 2017) - July 2017 - 11/21
New guaranteed, accurate and cheaperror estimate (EE)
Local problems: find qi such that∫ωi
qi ·∇vdΩ = R(φiv) ∀v ∈ V(ωi)
minimizing the local complementary energy∫ωi
qi · qidΩωi
φi
KEY POINT
Find a closed EXPLICIT solution for qi
A new equilibrated residual method (CMN 2017) - July 2017 - 11/21
New guaranteed, accurate and cheap EE
γ[1]
giγ[1]
γ[2]
giγ[2]
γ[3]
giγ[3]
γ[4]
giγ[4]
γ[5]
giγ[5]
γ[6]
giγ[6]
From star ωi to elements Ωk ⊂ ωi∫ωi
qi ·∇vdΩ = R(φiv)
The explicit solution is foundintroducing the linear tractions onthe edges of the star giγ[m]
wwfor every Ωk ⊂ ωi
Neumann BC (qik + φi∇uh) · nγk = σγkg
iγ
Divergence −∇ ·(qik + φi∇uh
)= φi(f − κ2uh)−∇uh ·∇φi
A new equilibrated residual method (CMN 2017) - July 2017 - 12/21
New guaranteed, accurate and cheap EE
γ[1]
giγ[1]
γ[2]
giγ[2]
γ[3]
giγ[3]
γ[4]
giγ[4]
γ[5]
giγ[5]
γ[6]
giγ[6]
From star ωi to elements Ωk ⊂ ωi∫ωi
qi ·∇vdΩ = R(φiv)
The explicit solution is foundintroducing the linear tractions onthe edges of the star giγ[m]
wwfor every Ωk ⊂ ωi
Neumann BC (qik + φi∇uh) · nγk = σγkg
iγ
Divergence −∇ ·(qik + φi∇uh
)= φi(f − κ2uh)−∇uh ·∇φi
A new equilibrated residual method (CMN 2017) - July 2017 - 12/21
New guaranteed, accurate and cheap EE
γ[1]
giγ[1]
γ[2]
giγ[2]
γ[3]
giγ[3]
γ[4]
giγ[4]
γ[5]
giγ[5]
γ[6]
giγ[6]
From star ωi to elements Ωk ⊂ ωi∫ωi
qi ·∇vdΩ = R(φiv)
The explicit solution is foundintroducing the linear tractions onthe edges of the star giγ[m]
wwfor every Ωk ⊂ ωi
Neumann BC (qik + φi∇uh) · nγk = σγkg
iγ
Divergence −∇ ·(qik + φi∇uh
)= φi(f − κ2uh)−∇uh ·∇φi
A new equilibrated residual method (CMN 2017) - July 2017 - 12/21
New guaranteed, accurate and cheap EE
γ[1]
giγ[1]
γ[2]
giγ[2]
γ[3]
giγ[3]
γ[4]
giγ[4]
γ[5]
giγ[5]
γ[6]
giγ[6]
From star ωi to elements Ωk ⊂ ωi∫ωi
qi ·∇vdΩ = R(φiv)
The explicit solution is foundintroducing the linear tractions onthe edges of the star giγ[m]
wwfor every Ωk ⊂ ωi
Neumann BC (qik + φi∇uh) · nγk = σγkg
iγ
Divergence −∇ ·(qik + φi∇uh
)= φi(f − κ2uh)−∇uh ·∇φi
A new equilibrated residual method (CMN 2017) - July 2017 - 12/21
New guaranteed, accurate and cheap EEStrong form of the elementary problems:
−∇ ·(qik + φi∇uh
)= φi( f − κ2uh )−∇uh ·∇φi in Ωk
qik · nγk = σγkg
iγ − φi∇uh · nγ
k := R|γ on ∂Ωk
Explicit solution of the elementary problems: qik = qiLk + qiCk
qiLk =1
2|Ωk|
3∑n=1
∑m=1m6=n
`[m] R|γ[m](x[n]) t[mn]λ[n]
qiCk =1
3
3∑n=1
3∑m=2m>n
λ[n]λ[m]t[nm]tT[nm] ∇vQ
A new equilibrated residual method (CMN 2017) - July 2017 - 13/21
New guaranteed, accurate and cheap EEStrong form of the elementary problems:
−∇ ·(qik + φi∇uh
)= φi( f − κ2uh )−∇uh ·∇φi in Ωk
qik · nγk = σγkg
iγ − φi∇uh · nγ
k := R|γ on ∂Ωk
Explicit solution: qik = qiLk + qiCk as long as∫Ωk
[φi(f − κ2uh
)−∇uh · ∇φi
]dΩ +
∑γ⊂∂Ωk
∫γ
σγkgiγ dΓ = 0,
Details can be found in
N. Pares, P. Dıez, A new equilibrated residual methodimproving accuracy and efficiency of flux-free error es-timates, CMAME, Volume 313, Pages 785-816 (2017)
Explicit solution of the elementary problems: qik = qiLk + qiCk
qiLk =1
2|Ωk|
3∑n=1
∑m=1m6=n
`[m] R|γ[m](x[n]) t[mn]λ[n]
qiCk =1
3
3∑n=1
3∑m=2m>n
λ[n]λ[m]t[nm]tT[nm] ∇vQ
A new equilibrated residual method (CMN 2017) - July 2017 - 13/21
New guaranteed, accurate and cheap EEStrong form of the elementary problems:
−∇ ·(qik + φi∇uh
)= φi( f − κ2uh )−∇uh ·∇φi in Ωk
qik · nγk = σγkg
iγ − φi∇uh · nγ
k := R|γ on ∂Ωk
Explicit solution of the elementary problems: qik = qiLk + qiCk
qiLk =1
2|Ωk|
3∑n=1
∑m=1m6=n
`[m] R|γ[m](x[n]) t[mn]λ[n]
qiCk =1
3
3∑n=1
3∑m=2m>n
λ[n]λ[m]t[nm]tT[nm] ∇vQ
x[1]
λ[1]
x[3]
x[2]
t[12] t[23]
t[31]
A new equilibrated residual method (CMN 2017) - July 2017 - 13/21
New guaranteed, accurate and cheap EEStrong form of the elementary problems:
−∇ ·(qik + φi∇uh
)= φi( f − κ2uh )−∇uh ·∇φi in Ωk
qik · nγk = σγkg
iγ − φi∇uh · nγ
k := R|γ on ∂Ωk
Explicit solution of the elementary problems: qik = qiLk + qiCk
qiLk =1
2|Ωk|
3∑n=1
∑m=1m6=n
`[m] R|γ[m](x[n]) t[mn]λ[n]
qiCk =1
3
3∑n=1
3∑m=2m>n
λ[n]λ[m]t[nm]tT[nm] ∇vQ
qiLk imposes the tractions on the element
qiCk is traction free
x[1]
λ[1]
x[3]
x[2]
t[12] t[23]
t[31]
A new equilibrated residual method (CMN 2017) - July 2017 - 13/21
New guaranteed, accurate and cheap EEStrong form of the elementary problems:
−∇ ·(qik + φi∇uh
)= φi( f − κ2uh )−∇uh ·∇φi in Ωk
qik · nγk = σγkg
iγ − φi∇uh · nγ
k := R|γ on ∂Ωk
Explicit solution of the elementary problems: qik = qiLk + qiCk
qiLk =1
2|Ωk|
3∑n=1
∑m=1m6=n
`[m] R|γ[m](x[n]) t[mn]λ[n]
qiCk =1
3
3∑n=1
3∑m=2m>n
λ[n]λ[m]t[nm]tT[nm] ∇vQ
vQ = 38φiF + 1
8(4F[1]λ[1] − F[2]λ[3] − F[3]λ[2])
qiCk imposes the divergence condition
F
x[1]
λ[1]
x[3]
x[2]
t[12] t[23]
t[31]
A new equilibrated residual method (CMN 2017) - July 2017 - 13/21
New guaranteed, accurate and cheap EE
LOCAL QUADRATIC CONSTRAINED
OPTIMIZATION PROBLEM:
find giγ[m] solution of
giγ[1]giγ[2]
giγ[3]
giγ[4]giγ[5]
giγ[6]
Minimize∑
Ωk⊂ωi
∫Ωk
qik(giγ) · qik(giγ)dΩ
giγ
Subject to
∫Ωk
[φi(f − κ2uh
)−∇uh · ∇φi
]dΩ
+∑γ⊂∂Ωk
∫γ
σγkgiγ dΓ = 0one restriction
per element
two dof per edge
A new equilibrated residual method (CMN 2017) - July 2017 - 14/21
Hybrid-flux vs. Explicit Flux-free
Hybrid-flux/equilibrated Explicit Flux-free
Min gγ − [[∇uh · n]]ave Min∑
Ωk⊂ωi
∫Ωk
qik(giγ) · qik(giγ)dΩ
gγ giγ
one dof per edge two dof per edge
+ one restriction per element in both cases
A new equilibrated residual method (CMN 2017) - July 2017 - 15/21
Constant Explicit Flux-free
γ[1]
giγ[1]
γ[2]
giγ[2]
γ[3]
giγ[3]
γ[4]
giγ[4]
γ[5]
giγ[5]
γ[6]
giγ[6]
two dof per edge
γ[1]
gi,cstγ[1]
γ[2]
gi,cstγ[2]
γ[3]
gi,cstγ[3]
γ[4]
gi,cstγ[4]
γ[5]
gi,cstγ[5]
γ[6]
gi,cstγ[6]
one dof per edge
giγ = φigi,cstγ
A new equilibrated residual method (CMN 2017) - July 2017 - 16/21
2D exampleUniformly forced square domain
−∆u = 1 in [−1, 1]2 with homogeneous Dirichlet BC
u(x, y) = 1−x22− 16
π3
+∞∑k=1odd
sin(kπ(1+x)/2)(sinh(kπ(1+y)/2)+sinh(kπ(1−y)/2))k3 sinh(kπ)
ρ = |||e|||ub/|||e||| ≈ 1 FLUX-FREE EQUILIBRATED
explicit
nel |||e||| ρst ρ ρq ρeq
8 0.34331271 1.00036 1.09131 1.01545 1.20880
32 0.27603795 1.04611 1.05288 1.03831 1.48894
128 0.15288301 1.04314 1.04621 1.03889 1.51749
512 0.07856757 1.04088 1.04470 1.03938 1.52104
2048 0.03955958 1.03948 1.04429 1.03962 1.51898
8192 0.01980831 1.03862 1.04420 1.03974 1.51641
32768 0.00990510 1.03813 1.04419 1.03982 1.51453
A new equilibrated residual method (CMN 2017) - July 2017 - 17/21
3D example3D diffusion problem with data oscillation
−∆u = f in [−1, 1]3 with Dirichlet BC
u(x, y, z) = e−20(x2+y2+z2)
102 103 104 105
Number of nodes (d.o.f.)
10-1
100
Energ
y norm
of th
e erro
r
||e|| adaptUB adapt||e|| unifUB unifN -1/3
Exact solution
Source term
A new equilibrated residual method (CMN 2017) - July 2017 - 18/21
3D example3D diffusion problem with data oscillation
−∆u = f in [−1, 1]3 with Dirichlet BC
u(x, y, z) = e−20(x2+y2+z2)
102 103 104 105
Number of nodes (d.o.f.)
1 1.051.11.31.5
2
3
4
Effec
tivitie
s of t
he es
timato
r
||e|| adaptUB adapt||e|| unifUB unif
Exact solution
Source term
A new equilibrated residual method (CMN 2017) - July 2017 - 18/21
3D example
3D diffusion problem with data oscillation
|||e|||2 ≤nel∑k=1
(‖q‖[L2(Ωk)]3 + hk
π‖f − Π1f‖L2(Ωk)
)2
dual error data oscillation
A new equilibrated residual method (CMN 2017) - July 2017 - 19/21
Conclusions
We have developed a new technique to computeguaranteed upper bounds for the energy norm of the error(which can also be used to compute bounds for QoI)
The proposed strategy may be seen as either:
(1) an improved cheap version of the flux-free estimate(2) a new more acurate hybrid-flux equilibrated EE
Alleviating the cost of the flux-free approach does notintroduce a significant difference on accuracy
The new equilibrated tractions yield sharper bounds thanthe original ones
A new equilibrated residual method (CMN 2017) - July 2017 - 20/21
A new equilibrated residual method:improving accuracy and efficiency offlux-free error estimates in two and
three dimensions
Nuria PARES and Pedro DIEZ
Laboratori de Calcul Numeric (LaCaN)Universitat Politecnica de Catalunya (Barcelona)
http://www-lacan.upc.edu
A new equilibrated residual method (ADMOS 2017) - July 2017 - 21/21